Looks like anyconnect web client won't run under linux/java 64.
Check out the source that is run during initialization:
function loadPackage ()
{
if (IsLinux64)
{
ToggleHelpButtonEnable(true);
// Magic state -70 == java applet weblaunch not supported, skip to manual download
Start(-70);
}
else
{
// Magic state -60 == go to the next installed package
Start(-60);
}
}